arX

iv:0

911.

2795

v3 [

astr

o-ph

.SR

] 2

Apr

201

0Accepted for publication in ApJ

Preprint typeset using LATEX style emulateapj v. 11/10/09

TOWARD A MAGNETOHYDRODYNAMIC THEORY OF THE STATIONARY ACCRETION SHOCKINSTABILITY:

TOY MODEL OF THE ADVECTIVE-ACOUSTIC CYCLE IN A MAGNETIZED FLOW

Jerome Guilet and Thierry Foglizzo

Laboratoire AIM, CEA/DSM-CNRS-Universite Paris Diderot, IRFU/Service d’Astrophysique,CEA-Saclay F-91191 Gif-sur-Yvette, France.

Accepted for publication in ApJ

ABSTRACT

The effect of a magnetic field on the linear phase of the advective-acoustic instability is investi-gated, as a first step toward a magnetohydrodynamic (MHD) theory of the stationary accretion shockinstability taking place during stellar core collapse. We study a toy model where the flow behind aplanar stationary accretion shock is adiabatically decelerated by an external potential. Two magneticfield geometries are considered: parallel or perpendicular to the shock. The entropyvorticity wave,which is simply advected in the unmagnetized limit, separates into five different waves: the entropyperturbations are advected, while the vorticity can propagate along the field lines through two Alfvenwaves and two slow magnetosonic waves. The two cycles existing in the unmagnetized limit, advec-tiveacoustic and purely acoustic, are replaced by up to six distinct MHD cycles. The phase differencesamong the cycles play an important role in determining the total cycle efficiency and hence the growthrate. Oscillations in the growth rate as a function of the magnetic field strength are due to this varyingphase shift. A vertical magnetic field hardly affects the cycle efficiency in the regime of super-Alfvenicaccretion that is considered. In contrast, we find that a horizontal magnetic field strongly increasesthe efficiencies of the vorticity cycles that bend the field lines, resulting in a significant increase of thegrowth rate if the different cycles are in phase. These magnetic effects are significant for large-scalemodes if the Alfven velocity is a sizable fraction of the flow velocity.Subject headings: instabilities — shock waves — magnetic fields—magnetohydrodynamics—

supernovae: general

1. INTRODUCTION

The delayed energy deposition by neutrinos maybe a viable explosion mechanism of massive stars(Bethe & Wilson 1985) if multidimensional hydrody-namical instabilities are taken into account during thefirst second after shock bounce, in the inner 200kmof the massive star (Buras et al. 2006; Marek & Janka2009). By capturing some gas in large scale convectivecells, these instabilities increase the exposure time ofthis gas to the heating by neutrinos (Marek & Janka2009; Murphy & Burrows 2008; Fernandez & Thompson2009a). The l = 1, 2 deformation of the shock seemsto be mainly due to the Standing Accretion ShockInstability (SASI), discovered by Blondin et al. (2003).This asymmetric instability may influence the kickand spin of the residual pulsar (Scheck et al. 2006;Blondin & Mezzacappa 2007), trigger the excitationof g-modes (Burrows et al. 2006) and the emissionof gravitational waves (Marek et al. 2009; Ott 2009).These potential consequences of SASI are based onnumerical simulations where the magnetic field isneglected, because the magnetic pressure is thoughtto be small compared to the thermal pressure ina typical core collapse. Other models of core col-lapse supernovae, in which magnetic forces play adynamical role, usually rely on a fast rotation ratefrom which the magnetic energy is extracted byfield winding and local instabilities (Akiyama et al.2003; Moiseenko et al. 2006; Shibata et al. 2006;

jerome.guilet@cea.fr

Burrows et al. 2007; Obergaulinger et al. 2009). Sur-prisingly, Endeve et al. (2008) argued that the turbulentflow associated with SASI may be able to amplify aninitially weak magnetic field to a dynamically significantstrength of 1015G, even if the progenitor is not rotating.This latter result, if confirmed, shows the importance ofunderstanding SASI in the context of MHD even if theprogenitor is weakly magnetized.As a first step in this direction, we investigate the

linear phase of SASI in a moderate magnetic field.The linear mechanism of SASI has been identified asan advective-acoustic cycle by Blondin et al. (2003);Ohnishi et al. (2006); Foglizzo et al. (2007); Scheck et al.(2008); Fernandez & Thompson (2009b). The interplayof acoustic waves and advected perturbations of en-tropy/vorticity has been illustrated using a simple toymodel by Foglizzo (2009) and Sato et al. (2009), in whichthe flow is planar and adiabatic. A similar cycle is ex-pected to exist in a MHD flow, but should be modi-fied by the magnetic field depending on its strength anddirection, and could potentially involve entropy waves,fast and slow magnetosonic waves, and Alfven waves.The local stability of an isolated fast MHD shock hasbeen established by Gardner & Kruskal (1964), and itsinteraction with MHD waves has been characterized byMcKenzie & Westphal (1970). However, the global sta-bility of the cycles resulting from the linear coupling ofthese MHD waves through a region of deceleration be-low the shock has never been studied. For this purposewe extend the toy model of Foglizzo (2009) to a magne-tized fluid, and investigate two possible orientations of

2 Guilet & Foglizzo

the magnetic field.It should be noted that the existence of SASI type

instabilities in a magnetized flow has been recently pro-posed by Camus et al. (2009) to explain the oscillationsof the termination shock in their MHD simulations of therelativistic pulsar wind in the Crab nebula. The presentstudy can thus be considered more generally as a firststep toward understanding the MHD extension of theclassical advective-acoustic cycle.In Sect. 2 we describe the stationary flow of our MHD

toy model and define the method of our linear analy-sis. The efficiencies of all existing cycles are calculatedin Sect. 3, and the growth rate of the resulting eigen-mode is computed in Sect. 4. The results are summarizedand discussed in Sect. 5. Most algebraic derivations havebeen separated from the main text for the sake of clarity,and can be found in the Appendices.

2. SET UP: THE MAGNETIC TOY MODEL

2.1. The stationary flow

In the toy model of the advective-acoustic instabilityintroduced by Foglizzo (2009), an ideal gas character-ized by an adiabatic index γ = 4/3 flows in the nega-tive z-direction. This planar adiabatic flow is deceleratedthrough a stationary shock at z = zsh, with an incidentMach number M1. It is further decelerated over a dis-tance H∇ < H in the vicinity of z∇ ≡ zsh − H , by adecrease ∆Φ of the external potential :

Φ (z) ≡∆Φ

2

[

tanh

(

z − z∇H∇/2

)

+ 1

]

. (1)

We add to this flow a magnetic field (Bx, 0, Bz) that isoriented either vertically (Bx = 0), or horizontally (Bz =0), as illustrated in Fig. 1. This magnetic field does notdepend on x or y.Let us denote by P the gas pressure, ρ the density,

c ≡√

γP/ρ the sound speed, and v the fluid velocity.S ≡ [log ((P/Psh) / (ρ/ρsh)

γ)] / (γ − 1) is a dimensionless

measure of the entropy. vAi ≡√

B2i /µ0ρ is the Alfven

speed along the direction i = x or z .The stationary flow is governed by the conservation of

the mass flux ρvz , the energy density v2z/2+c2/ (γ − 1)+v2Ax+Φ, and the entropy S. The conservation of Bz andBx/ρ follows respectively from the conservation of themagnetic flux and the frozen-in field lines in ideal MHD.A generalization of the Rankine-Hugoniot jump con-

ditions to MHD characterizes the post-shock flow as afunction of the upstream conditions (Eqs. (A1-A2) in Ap-pendix A). The additional pressure due to a horizontalmagnetic field decreases the compression at the shockand increases the post shock Mach number by less than10% if vAsh < vsh.The flow below the adiabatic region of deceleration is

characterized by the size ∆Φ of the potential jump, orequivalently by the ratio of sound speed cin/cout, using

Fig. 1.— Schematic view of the toy model. A magnetic field thatis either vertical or horizontal is added to the hydrodynamical toymodel of Foglizzo (2009). Six waves propagate from the shock tothe inhomogeneous region near z∇ (an entropy wave S, two Alfvenwaves a±, two slow waves s± and a fast wave f+) and couple to theupward propagating fast wave f−. The linear coupling between awave i and the wave f− is described by the coupling coefficientsQi

∇ and Qish

defined in Sec. 2.2

the conservation of mass, energy and magnetic fluxes:

Bxin

Bxout=

(

cincout

)2

γ−1

, (2)

vinvout

=

(

cincout

)− 2γ−1

, (3)

Min

Mout=

(

cincout

)−γ+1

γ−1

, (4)

∆Φ=

[

v2

2+ v2A +

c2

γ − 1

]out

in

. (5)

Unless otherwise specified, we use the values of the pa-rameters c2in/c

2out = 0.75, H∇/H = 10−3, M1 = 5 when

studying the effect of the magnetic field. This choiceimplies an increase of the height of the potential jumpwith the field strength, which becomes significant forvAxsh/vsh > 1− 1.5, i.e. when the magnetic pressure be-comes comparable with the thermal pressure (Fig. 2). Al-ternatively, one could choose to keep the potential jumpconstant. This choice of parameterization is unimportantin the present study, which is focussed on the regime ofweak magnetic pressure.

2.2. Linear perturbations

MHD SASI 3

Fig. 2.— Effect of the magnetic field on the stationary flow.Upper panel: Ratio of magnetic pressure to thermal pressure justbelow the shock (full line) and below the potential jump (dashedline). Bottom panel: Change of the potential jump if the compres-sion is kept constant with c2

in/c2out = 0.75. The potential jump is

increased by ∼ 20% at vA = 1.

The 1-D stationary flow is perturbed in 3-D, usingFourier transforms in time and in the directions x and ywhere the boundary conditions are periodic. The struc-ture of the perturbation of a physical quantity f is thus:δf(z) × exp i (kxx+ kyy − ωt). For a given horizontalsize of the computational domain Lx and Ly (we choseLx = Ly = 4H), the horizontal wavenumbers take dis-crete values ki ≡ 2πni/Li, where i = x, y and ni is thenumber of horizontal wavelengths in the i-direction. Weimpose a leaking boundary condition below the potentialjump (i.e. no wave propagating upward from below), andleave the supersonic flow unperturbed ahead of the shock.The corresponding equations are detailed in AppendicesA and B.In each uniform section of the flow, perturbations can

be decomposed into 7 waves: two fast magnetosonic, twoslow magnetosonic, two Alfven and one entropy wave.If the shock is a fast MHD shock, only the fast mag-netosonic wave is able to reach it from below. We donot consider a slow MHD shock because it would re-quire Alfven waves to be able to propagate upstreamof the shock (which would correspond to an unrealisti-cally strong magnetic field), nor transAlfvenic disconti-nuities, which have been shown to be non evolutionary(Syrovatskii 1959). As a consequence, we always considera situation where, due to advection, 6 of the 7 waves arepropagating downward in the post-shock region (Fig. 1).A global mode can be decomposed into 6 cycles, asso-

ciated with each of the six waves propagating downward,while the upward propagation always corresponds to afast magnetosonic wave. Waves are linearly coupled atthe shock and in the region of flow gradients around z∇.Following Foglizzo (2009), the coupling efficiency Qi

∇ de-

scribes the ratio of amplitudes of a perturbation δA ofa physical quantity A which is propagated from zsh toz∇, generates a fast magnetosonic wave in the region ofgradients, which propagates back to the shock:

Qi∇≡

δAf−sh

δAish

, (6)

i = s±, f+, a±, S,

where the symbol + (−) is used for waves propagatingdownward (upward) in the frame advected with the fluid.The superscript s± stands for slow magnetosonic waves,f± for the fast ones, a± for the Alfven waves and S forthe entropy wave.The coupling efficiency at the shock Qi

sh measures thecreation of a wave i when the fast magnetosonic wave f−perturbs the shock:

Qish ≡

δAish

δAf−sh

. (7)

The choice of the physical quantity A is driven by thesimplicity of the resulting equations (see Appendices Aand B). The total cycle efficiency Qi, defined as follows,does not depend on this choice:

Qi ≡ Qi∇Q

ish. (8)

We define Qtot as the sum of all the cycles constants,and Qvort ≡ Qs+ +Qs− +Qa+ +Qa− is restricted to thecycles involving vorticity.It is instructive to study how these six cycles relate to

the two hydrodynamical cycles, by considering the weakfield limit. Since the acoustic cycle straightforwardly be-comes the fast magnetosonic cycle, its reflection efficiencyis noted R instead of Qf+. The advective-acoustic cyclesplits between an entropy-fast cycle where the entropywave is still advected, and four Alfven-fast, slow-fast cy-cles that contain the vorticity, which is no longer pas-sively advected. In the setup we consider, the entropycycle typically has an efficiency of QS ∼ 1 while the vor-ticity cycle has an efficiency of Qvort ∼ 0.3 in the hydro-dynamical limit. How is vorticity distributed betweenthe Alfven and slow waves ? The distinction betweenslow and Alfven waves at the shock can be anticipatedby remembering that the velocity and magnetic field per-turbations of Alfven waves are along the direction k×B,while that of slow waves is in the plane (k,B) (or equiv-alently the vorticity of slow waves is along k ×B, whilethat of Alfven waves is in the plane (k,B)). This ques-tion is discussed in the next two subsections, dependingon the orientation of the magnetic field. In the weakfield limit, the efficiencies associated with Alfven andslow waves are independent of the direction of propa-gation (Qa+ ∼ Qa− and Qs+ ∼ Qs−), because both theAlfven and slow waves are then simply advected and the(±) components are undistinguishable.

2.3. Vertical magnetic field

A vertical magnetic field does not influence the station-ary flow as the fluid flows along the field lines without ex-periencing any magnetic force. It does however influencethe evolution of perturbations that involve some trans-verse motion, in particular enabling vorticity to propa-gate. The equations governing the perturbations are de-

4 Guilet & Foglizzo

scribed in the Appendix B. With the magnetic field in thez-direction, the directions x and y are equivalent. Withno loss of generality we choose the x-direction parallelto the transverse wavenumber (ky = 0) and solve a 2Dproblem in the plane (x, z). The boundary conditionsat the shock (Eqs (B13-B14)) impose δvy = δBy = 0.The only waves along this direction would be the Alfvenwaves, which displacement is along k × B (i.e. along ywith our choice of axis). As a consequence, the shock os-cillations cannot create Alfven waves, and there are only4 cycles in this particular field geometry. Note that theabsence of Alfven waves is not a consequence of choosingky = 0.In a stationary accretion flow in uniform vertical mag-

netic field, the ratio of the flow velocity to the Alfvenvelocity decreases downward, scaling like v/vA ∝ ρ−1/2.The transition from a superAlfvenic flow to a subAlfvenicflow is named the Alfven surface. Perturbations cannotbe treated in the framework of ideal MHD in the vicin-ity of this surface because their wavelength becomes in-finitely small. Alfven waves can accumulate and be am-plified there (Williams 1975). For the sake of simplicity,we restrict the present study to weak enough magneticfields so that the flow is superAlfvenic everywhere. Givenour choice of parameters, the superAlfvenic condition re-quires vAzsh/vsh < 0.68. The study of transAlfvenic flowsis the subject of a separate study (Guilet et al., in prep.).

2.4. Horizontal magnetic field

The equations governing the perturbations in a hori-zontal magnetic field are described in the Appendix A.In general, both Alfven and slow waves are created at

the shock. They propagate mainly along the horizontalmagnetic field lines (slow waves also propagate slightlyalong the perpendicular direction).The two particular cases kx = 0 and ky = 0 are simpler

because the evolution of perturbations is then planar:(i) if ky = 0, the boundary conditions at the shock

impose δvy = δBy = 0 while vx and Bx are perturbedby the shock oscillations. As k × B is the y direction,the shock oscillations do not create Alfven waves but docreate slow waves.(ii) if kx = 0 (k⊥B), the magnetic field lines are not

bent and the only magnetic force is the magnetic pressurewhich adds up to the thermal pressure in the plane (y, z)perpendicular to the field. The evolution of perturbationis thus similar to the hydrodynamical limit, with a mod-ified pressure. Vorticity is advected by the flow togetherwith entropy. This vorticity wave should be viewed asan Alfven wave rather than a slow wave, because it is inthe direction of the magnetic field.

3. THE COUPLING EFFICIENCY IN THE PRESENCE OF AMAGNETIC FIELD

In this section we study the effect of the magnetic fieldon the coupling efficiencies, which are computed by in-tegrating numerically the differential system governingthe perturbations, and using the boundary conditionsgiven in the appendices. We consider plane waves witha real frequency which coincide with the real part of theeigenfrequency of a given eigenmode. Growth rates arediscussed in Sect. 4.

3.1. Vertical magnetic field

Fig. 3.— Upper panel: Efficiency of the slow cycles Qs± as afunction of the height of the potential jump H∇ in the presenceof a vertical magnetic field. The grey line represents the efficiencyin the absence of a magnetic field, the black lines correspond tovAzsh = 0.5vsh. The dashed line is the (-) wave, the full line the(+) wave. The wave propagating up (Qs−) is more affected bythe cutoff than the wave propagating down (Qs+). Bottom panel:The cycle efficiencies are normalized by Q0 defined as the limit ofthis efficiency when the potential jump is compact. The cutt offhappens when kzH∇ & 1− 2, where kz is the vertical wavenumberof the slow wave involved in the cycle.

By solving the evolution of perturbations described inAppendix A, we remark that the influence of a verticalmagnetic field on the individual coupling constants of thedifferent cycles is moderate (< 20%) in the long wave-length limit, where the compact approximation is valid.The efficiencies of the two slow cycles at shorter wave-length differ in a way which can be understood in termsof the cutoff described by Foglizzo (2009). In the weakfield limit, the vertical wavenumber of slow waves can beapproximated by:

ks±z ≃ω

vz ∓ vAz. (9)

The slow wave propagating against the stream (-) has ashorter wavelength than the wave propagating with thestream (+). It is thus more sensitive to the cutoff inducedby the finite scaleheight of the gradients (Fig. 3, upperpanel). As in the hydrodynamic case, the cutoff takesplace for kzH∇ ∼ 1 (Fig. 3, bottom panel).

3.2. Horizontal magnetic field

3.2.1. Straight field lines (k ⊥ B)

We observe the presence of two slow cycles, whichwere absent in the non-magnetic case (Fig. 4). Theseslow waves taken together (+ and - are equivalent hereas these slow waves do not propagate) do not containany velocity or vorticity perturbation: they are simplya perturbation of the magnetic field strength and den-sity in such a proportion that the total pressure is un-perturbed. As their efficiency depends weakly on thetransverse wavenumber, we focus on the simplest case

MHD SASI 5

Fig. 4.— Apparition of a slow cycle in the presence of a horizontalmagnetic field and k⊥B. This cycle carries no vorticity. It is only aperturbation of the magnetic field and density in such a proportionthat the total pressure is not perturbed.

of ny = 0. In this case, the presence of the slow wavescan be explained using the conservation of magnetic flux,which perturbation δA ≡ δBx/B − δρ/ρ should vanishat the shock (Eqs. (A29)).While fast and Alfven waves do not perturb the mag-

netic flux, the entropy wave is responsible for a perturba-tion δAS = (γ − 1) δS/γ. Slow waves have to be createdat the shock in order to keep δAsh = 0.In the deceleration region, the slow waves necessarily

disturb the pressure equilibrium and create an acous-tic feedback, in order to satisfy the conservation of boththe mass flux and the magnetic flux across the poten-tial jump. This acoustic generation is very similar to theacoustic feedback produced by the deceleration of the en-tropy wave, required by the conservation of entropy andmass flux across the potential jump.The effect of the magnetic field on other cycles is only

minor and depends on the parameterization of the po-tential jump. For example the efficiency of the entropycycle is slightly increased if the compression is kept con-stant when the magnetic field is varied, while it is slightlydecreased if the height of the potential jump is kept con-stant.

3.2.2. Bending the field lines: amplification of the vorticalcycles

If k ‖ B, the slow cycles carrying the vorticity arestrongly amplified by the presence of a horizontal mag-netic field (Fig. 5, upper panel). The amplification is al-most linear with the magnetic field strength and reachesa factor ∼ 5 at vAxsh = vsh. Note that this effect is moreimportant than the one described in Sect. 3.2.1, whichefficiency only reached Qs± ∼ 0.1 at vAxsh = vsh.When k is oblique with respect to B, the vorticity is

distributed in both the Alfven and the slow waves. TheAlfven cycles are also amplified, although to a lesser ex-tent than the slow cycles (Fig. 5, bottom panel). Whenvarying the angle between k and B, the total vortical cy-cle efficiency (the sum of the four slow and Alfven cycles)increases with kz/k.Although we do not provide an analytical description

of this amplification, one can gain insights into its natureby first remarking that this effect is not restricted to slowwaves and hence is not due to the compressional nature ofthe slow waves. Second, this amplification appears only

Fig. 5.— Amplification of the vorticity cycles in the presence ofa horizontal magnetic field. Upper panel: k ‖ B (nx = 1, Lx = 4),only slow waves are created at the shock. Bottom panel: k obliquewith respect to B (nx = ny = 1, and Lx = Ly = 4

√2). Both slow

and Alfven waves are created at the shock, and amplified by thepresence of the horizontal magnetic field.

when the field lines are bent by the perturbations. Oneshould however note that a vertical magnetic field bentby perturbations does not lead to the same amplification.

4. BUILDING OF A GLOBAL MODE BY ADDING UPCYCLES

4.1. Method

The cycle efficiencies can be calculated using a com-plex frequency ω = ωr + iωi, where ωr is the oscillationfrequency and ωi the growth rate. Generalizing the anal-ysis of Foglizzo (2009), the complex eigenfrequency ω ofa global mode obeys the following relation:

Qtot ≡ QS +Qs+ +Qs− +Qa+ +Qa− +R = 1. (10)

The cycle efficiencies computed in Sect. 3 with a realfrequency can be used to characterize marginal stability(|Qtot| = 1). |Qtot| < 1 indicates a stable mode, and|Qtot| > 1 an unstable one.An estimate of the growth rate associated with a single

cycle in the limit ωi ≪ ωr has been given in Foglizzo(2009):

ωi ≃1

τcyclelog |Q(ωr)|, (11)

where τcycle is the cycle timescale. In the case of anadvective-acoustic cycle,

τcycle = τaacµ+Msh

µ (1 +Msh)(12)

6 Guilet & Foglizzo

where µ is defined by µ2 ≡ 1 − k2xc2(

1−M2)

/ω2 andτaac is the radial advective-acoustic time:

τaac ≡ H/(|vsh| (1−Msh)). (13)

The situation is more complicated if several cycles withdifferent timescales are involved in the instability. Onemay however propose an approximate relation:

ωi ≃1

τefflog |Qtot(ωr)|, (14)

where τeff is an effective timescale that would be an av-erage of the different cycle timescales. In the next twosubsections we use Eq. (12) as a proxy for τeff and showthat this is a good approximation.

4.2. Vertical magnetic field

Fig. 6 shows the evolution of the most unstable nx = 1and nx = 8 modes with a vertical magnetic field. Theestimate of the growth rate using Qtot (Eq. (14)) is inexcellent agreement with the eigenvalue if nx = 8, whilethe agreement is only reasonable if nx = 1 (in which casethe condition ωi ≪ ωr is less justified). This justifiesa posteriori our estimate of the effective timescale τeff ,and indicates that it is weakly affected by the magneticfield. Some effect of the magnetic field could have beenexpected since the vorticity is able to propagate throughslow waves, but the (s-) wave is decelerated while the(s+) wave is accelerated, so that both effects on τeff can-cel each other to first-order. Furthermore, the most effi-cient cycle in our toy model is the entropic-acoustic cycleand at the magnetic field strength considered, the prop-agation speed of the fast magnetosonic waves is close tothe acoustic one.The growth rate as a function of the magnetic field

strength shows oscillations (Fig. 6, upper panel), whichare reminiscent of the oscillations in the SASI eigenspec-trum observed by Foglizzo et al. (2007) (Fig. 7). Thesewere interpreted as the consequence of a purely acous-tic cycle interacting either constructively or destructivelywith the advective-acoustic cycle. Similarly the presentoscillations can be interpreted as the constructive or de-structive effect of the vortical cycles (the two slow cycles).The phase of their cycle efficiencies Qs+ and Qs− varieswith the vertical wavevector of the slow waves as ks±z H(neglecting a variation of the phase shift during the cou-pling process). Equation (9) shows that the phases ofQs+ and Qs− vary in opposite directions. This results inoscillations in the value of Qvort as well as Qtot (Fig. 6).As vA approaches vz , the phase of Qs− varies faster andfaster, which can recognized in the faster and faster os-cillations in Fig. 6. The slower phase variation of Qs+ isresponsible for the slower background oscillations in thegrowth rate of the nx = 8 mode.Using a first-order expansion of Eq. (9), an estimate

of the first minimum of the growth rate can be deducedfrom the criterion (kz − kz0)H = π:

vAzsh

vsh∼

πvshωH

∼1

2h (1−Msh), (15)

where the second estimate is obtained by approximat-ing the h-th harmonics frequency by ω = 2hπ/τaac.This gives vAzsh/vsh ∼ 0.83 for the mode nx = 1, andvAzsh/vsh ∼ 0.10 for nx = 8.

Fig. 6.— Effect of a vertical magnetic field on the modes nx =1 (black) and nx = 8 (grey). Upper panel: Growth rate (fullline) as a function of the vertical magnetic field strength. Thequantity log (Qtot) /τ is overplotted with dotted lines. Bottompanel: Efficiency of the vorticity cycles Qvor. The oscillations aredue to interferences between the downward propagating (+) andupward propagating (-) slow waves.

Because the entropy and the vorticity cycles areroughly in phase in the non-magnetic limit, the phasevariations due to the magnetic field induce an overall de-crease of Qtot and ωi (Figs. 6 and 7). For any strength ofthe magnetic field, some modes benefit from a construc-tive interference of the different cycles and are as unsta-ble as in the absence of a magnetic field (e.g. nx = 5, 10in Fig. 7). This phase effect thus never stabilizes com-pletely the compact toy model (H∇ = 0), but is responsi-ble for an irregular eigenspectrum and lowers the averagegrowth rate. If the deceleration region is sufficiently ex-tended vertically, the cutoff effect verified in Sect. 3.1 canstabilize high frequency modes. A strong enough mag-netic field may thus be able to completely stabilize theflow if the frequency of the constructive cycles exceedsthe frequency cutoff induced by the finite size of the de-celeration region.

4.3. Horizontal magnetic field

Similarly to the case of a vertical magnetic field, thegrowth rate is well described by Eq. (14) (Fig. 8). Theoscillations are due to the propagation of vorticity alongmagnetic field lines, which changes the vertical structureof the slow and Alfven waves as follows:

ks±z ≃ ka±z =ω

v±

kxvAv

(16)

MHD SASI 7

Fig. 7.— Effect of a vertical magnetic field on the eigenspectrumof the compact toy model (H∇ = 0). The most unstable modeis shown for each nx from 1 to 10. White diamonds are withoutmagnetic field, black diamonds with vAzsh = 0.5vsh.

Transverse slow modes are most affected by the magneticfield. The magnetic field strength of the first minimum inthe growth rate oscillations, due to the destructive effectof the slow or Alfven cycles on the entropic cycle, can beestimated as:

vAxsh

vsh∼

π

kxH=

Lx

2nxH(17)

This gives vAxsh/vsh ∼ 2 if nx = 1, and vAxsh/vsh ∼ 0.25if nx = 8.Furthermore, due to the amplification of the vortical

cycles described in Sec. 3.2.2, the growth rate at theoscillation maximum increases with the magnetic fieldstrength (Fig. 8). As an example, the mode nx = 8grows four times faster at vAxsh = vsh than at vAxsh = 0.The eigenspectrum is more irregular than the hydrody-namical one, due to the interferences between the dif-ferent cycles, and shows higher maximum growth rates:ωiτaac ∼ 0.8 if vAxsh = vsh, compared to ωiτaac ∼ 0.4 ifvAxsh = 0 (Fig. 9).

5. DISCUSSION AND CONCLUSION

The main results of this study can be summarized asfollows:

1. In the presence of a magnetic field, the advective-acoustic cycle splits in up to 5 different cycles. Theacoustic wave becomes a fast magnetosonic wave,while the entropy-vorticity splits into an entropywave, 2 Alfven waves and 2 slow magnetosonicwaves.

2. The acoustic cycle becomes a fast magnetosonic cy-cle. Its efficiency is not significantly affected by thepresence of a magnetic field in any of the configu-rations considered in this paper.

3. The propagation of the vorticity through slow andAlfven waves leads to a phase difference betweenthe different cycles, which interact either construc-tively or destructively depending on the mode con-sidered. The consequence of these interferences isa more irregular eigenspectrum.

Fig. 8.— Same as Fig. 6 but with a horizontal magnetic field.The modes presented here have k ‖ B: nx = 1− 8, ny = 0.

Fig. 9.— Effect of a horizontal magnetic field on the eigenspec-trum. The most unstable mode is shown for nx varying from 1 to10, and ny = 0. White diamonds are without magnetic field, blackdiamonds with vAxsh = vsh.

4. In the superAlfvenic regime that we investigate, avertical magnetic field hardly changes the couplingefficiencies in the compact toy model. It only af-fects the cutoff associated with the size H∇ of thedeceleration region by changing the vertical struc-ture of the slow waves.

5. The vortical cycles are strongly amplified by a hor-izontal magnetic field if the field lines are bent

8 Guilet & Foglizzo

(k.B 6= 0). Both the Alfven cycles and the slowmagnetosonic cycles are amplified. This amplifi-cation of the vortical cycles leads to faster growthrates of the modes with nx 6= 0.

These results confirm that the mechanism of theadvective-acoustic cycle can be generalized to a magne-tized environment, such as core collapse supernovae orpossibly the termination shock of pulsar wind nebulae.In order to estimate the significance of these results forSASI during core collapse, we can estimate the magneticfield strength needed to affect the advective-acoustic cy-cle. The desynchronization effect studied in Sects. 4.2and 4.3 requires vAsh ∼ vsh for low frequency, low nx

modes. Eq. (17) can be extrapolated to a spherical corecollapse as follows:

vAv

∼Rsh

2√

l(l+ 1)H. (18)

For the dominant l = 1 − 2 modes, with Rsh = 110kmand H = 50km, the desynchronisation of the cycles isexpected for vA/v ∼ 0.4− 0.8 .The magnetic field strength at which the amplifica-

tion of the vortical cycles becomes significant is typicallyvAsh ∼ (0.5− 1)× vsh in our toy model. A more detailedstudy should determine whether this amplification takesplace at the shock or in the region of deceleration.Interestingly the magnetic effects described in this pa-

per seem to be significant when the Alfven speed is com-parable with the advection speed rather than when themagnetic pressure is comparable with the thermal pres-sure. These two criteria are related through the Machnumber M:

Pmag

Pth=

γ

2M2

(vAv

)2

. (19)

In the subsonic flow the pressure ratio is thus smallerthan vA/v, by a factor ∼ 0.06 at the shock if Msh ∼ 0.3,and as low as 10−3 if M ∼ 0.05 at the coupling radius.This raises the possibility that the magnetic field mayaffect the growth of SASI through the magnetic tension,even if the field is so weak that the magnetic pressure isnegligible.The strength of the magnetic field present in the iron

core of a star before the collapse is very uncertain. Thebest estimate to date is: Bφ ∼ 5.109G and Br ∼ 106G(Heger et al. 2005), which is too weak to have any effecton SASI: indeed, flux conservation during the collapse(B/ρ2/3 = cste, e.g. Shibata et al. (2006)) would leadto vAsh/vsh ∼ 0.001 at the shock (rsh ∼ 150km, ρsh ∼109g.cm−3), and vA/v ∼ 0.025 at the proto-neutron starsurface (rPNS ∼ 50km, ρPNS ∼ 1011g.cm−3).A reference Alfven speed near the surface of a strongly

magnetized proto-neutron star can be estimated by as-suming that the fossil magnetic field is strong enoughto give birth to a magnetar (B ∼ 5.1015G) by simple

conservation of the magnetic flux :

vAv

∼ 0.93×BNS

5.1015G

(

4.1014g.cm−3

ρNS

)2/3( r

50km

)2

(

ρ

1011g.cm−3

)7/60.3M⊙.s

−1

M. (20)

Such a field could be enough to affect SASI significantlyif the amplification of the vortical cycle takes place atthe coupling radius, which is slightly above the proto-neutron star surface (Scheck et al. 2008). Conversely, ifthe amplification takes place at the shock, even a fossilmagnetic field corresponding to magnetar strength wouldhave a negligible effect on SASI (vAsh/vsh ∼ 0.027). Notehowever that even stronger magnetic fields are sometimesconsidered in the literature, with Alfven waves propagat-ing above the shock (Suzuki et al. 2008).Given the simplicity of our toy model, we cannot ex-

pect it to capture more than the first-order effects of themagnetic field on the advective-acoustic cycle. More ac-curate estimates should consider a more realistic setupthat includes radial convergence and non-adiabatic ef-fects, which are expected to affect the relative impor-tance of the entropy and vorticity cycles. These effectscould be estimated in the cylindrical geometry used byYamasaki & Foglizzo (2008).The role of the Alfven surface has not been considered

in the present study, because the region of acoustic feed-back was assumed to be superAlfvenic for the sake ofsimplicity. If the magnetic field were vertical and strongenough, the magnetic extension of the advective-acousticcycle to MHD cycles would have to take into account thepossible coupling processes taking place at the Alfvensurface. The physics of transAlfvenic accretion flows willbe considered in a forthcoming study (Guilet, Fromang& Foglizzo, in prep.).The amplification of the magnetic field observed in the

numerical simulations by Endeve et al. (2008) seems totake place in the nonlinear regime of the MHD-SASI in-stability, and is thus beyond the scope of the presentlinear study. The topology of the magnetic field consid-ered by Endeve et al. (2008) is initially vertical (a splitmonopole), for which our analysis did not reveal anysignificant magnetic destabilization. Based on the cur-rent understanding of the nonlinear saturation processof SASI by parasitic instabilities (Guilet et al. 2009), wewould rather anticipate a larger saturation amplitude ofSASI if the field were horizontal near the shock, due to i)the higher growth rate of the cycles involving the bend-ing of field lines by vorticity perturbations, ii) the re-sistance of magnetic tension to the development of par-asitic instabilities such as Rayleigh-Taylor and Kelvin-Helmholtz instabilities. As noted in Guilet et al. (2009)however, the Rayleigh-Taylor instability may still be ableto develop as a parasitic instability in the direction per-pendicular to the magnetic field, without bending thefield lines. Besides, the presence of other magneticallyinduced parasitic instabilities cannot be ruled out. Al-together, these qualitative statements are insufficient toexplain the growth of the magnetic energy observed byEndeve et al. (2008).

J.G. is thankful to the Physics Department of the Uni-versity of Central Florida for its hospitality. This workhas been partially funded by the Vortexplosion projectANR-06-JCJC-0119. The authors are grateful to theanonymous referee for his comments that improved theclarity of the paper.

MHD SASI 9

APPENDIX

HORIZONTAL MAGNETIC FIELD

Stationary flow

Denoting by MA1 = vA1/c1 the ratio of the Alfven speed and the sound speed ahead of the shock, the postshockMach number is influenced by a horizontal magnetic field as follows:

M21

M2sh

= χ2

[

1 + (γ − 1)

(

M21

2+M2

A1

)]

− (γ − 1)M2A1χ

3 − (γ − 1)M2

1

2, (A1)

where χ = ρsh/ρ1 is the compression factor, given by:

χ =

[

(

1 + γ−12 M2

1 +γ2M

2A1

)2+ (2− γ) (γ + 1)M2

1M2A1

]12

−(

1 + γ−12 M2

1 +γ2M

2A1

)

(2− γ)M2A1

. (A2)

Definition of the variables

The 7 variables defined in Eqs. (A7-A9) are chosen to describe the perturbations. This choice is guided by the questfor the simplest possible differential system in which the stationary flow gradients do not appear. These variables havethe advantage of being conserved through a compact potential jump. δA is the perturbation of B/ρ, which is conservedin the stationary flow due to magnetic flux conservation. δf is the perturbation of the Bernoulli invariant and δSis the entropy perturbation. δEx and δEy are the perturbation of the transverse electric field along the directions xand y, normalized by the stationary electric field Ey = −vB. δK1 describes the velocity along the y direction and isconserved in the absence of magnetic field (Yamasaki & Foglizzo 2008).

δA≡δBx

B−

δρ

ρ, (A3)

δf ≡ vδvz +2cδc

γ − 1+ v2A

(

2δBx

B−

δρ

ρ

)

, (A4)

δEx≡−δBy

B, (A5)

¯δvx≡ δvx −v2Av

δBz

B, (A6)

δEy ≡δBx

B+

δvzv

, (A7)

δK1≡ iωδvy − ikyδf, (A8)

δS≡1

γ − 1

(

δP

P− γ

δρ

ρ

)

. (A9)

The usual physical quantities can be expressed as a function of the new variables through the following set ofequations:

δBx

B= δEy −

δvzv

, (A10)

δBy

B=−δEx, (A11)

δBz

B=−

v

ω(kxδEy − kyδEx) , (A12)

δvx= ¯δvx −v2Aω

(kxδEy − kyδEx) , (A13)

δvy =1

ω(kyδf − iδK1) , (A14)

δvzv

=1

1 +M2A −M2

[

−δf

c2+ δS +

(

1 +M2A

)

δEy +(

M2A − 1

)

δA

]

, (A15)

δρ

ρ= δEy − δA−

δvzv

, (A16)

δc2

c2=(γ − 1)

[

δS + δEy − δA−δvzv

]

, (A17)

δwx=1

vr

[

δK1 + iky

(

c2δS

γ+ v2AδA

)

− v2AikxδEx

]

, (A18)

10 Guilet & Foglizzo

where wx is the vorticity along the x direction.

Differential system

The differential system governing the evolution of the perturbations in a horizontal magnetic field is:

∂δA

∂z=

iω

vδA+

ikxv

[

¯δvx −v2Aω

(kxδEy − kyδEx)

]

, (A19)

∂δf

∂z=

iω

v

(

v2δvzv

+ c2δS

γ+ v2AδA

)

+ikxv

2A

ωv

[

ω ¯δvx −(

v2 + v2A)

(kxδEy − kyδEx)]

, (A20)

∂δEx

∂z=

iω

vδEx +

ikxωv

(iδK1 − kyδf) , (A21)

∂ ¯δvx∂z

=iω

v¯δvx +

ikxv

[

(

c2 − v2A) δvz

v+ c2 (δA− δEy) + (1− γ) c2

δS

γ

]

, (A22)

∂δEy

∂z=

iω

v

(

δEy −δvzv

)

+ikyωv

(iδK1 − kyδf) , (A23)

∂δK1

∂z=

iω

vδK1 −

kxv2A

ωv

[

−kyω ¯δvx −(

k2y(

v2 + v2A)

+ ω2)

δEx + kxky(

v2 + v2A)

δEy

]

, (A24)

∂δS

∂z=

iω

vδS. (A25)

(A26)

Wave decomposition in a uniform flow

TABLE 1

Decomposition of the perturbations into waves: horizontal magnetic field

Fast ± Slow ±δA − k2

xc2

ω2c

δρρ

− kxcωc

δvxc

δf

[

kzvω2c−k2

xc2

ωc

(

k2z+k2

y

) + c2 + v2A

(

1− 2k2xc2

ω2c

)

]

δρρ

[

kzvkxc

ω2c−k2

xc2

k2z+k2

y

+ ωc

kxc+

v2A

c

(

ωc

kx− 2kxc2

ωc

)

]

δvxc

δEx kykxω2c−k2

xc2

ω2c

(

k2z+k2

y

)

δρρ

kyω2c−k2

xc2

cωc

(

k2z+k2

y

)

δvxc

¯δvx

(

kxc2

ωc+

v2Akxkz

v

ω2c−k2

xc2

ω2c

(

k2z+k2

y

)

)

δρρ

(

c+v2Akz

cv

ω2c−k2

xc2

ωc

(

k2z+k2

y

)

)

δvxc

δEy

(

1− k2xc2

ω2c

+ kz

v

ω2c−k2

xc2

ωc

(

k2z+k2

y

)

)

δρρ

(

ωc

kxc− kxc

ωc+ kz

kxvc

ω2c−k2

xc2

k2z+k2

y

)

δvxc

δK1 iky

[

ω2c−k2

xc2

k2z+k2

y

− c2 − v2A

(

1− 2k2xc2

ω2c

)]

δρρ

iky

[

ωc

kxc

ω2c−k2

xc2

k2z+k2

y

− ωc

kxc− v2

A

c

(

ωc

kx− 2kxc2

ωc

)

]

δvxc

δS 0 0

kz ∼ ωc

M∓µ

1−M2 ± v2A

2c2

(

k2y+k

±2z0

)

c

µω∼ ω

v

[

1± kxvAω

(

1− v2A

2c2

k2y+ω

2

v2

k2x+k2

y+ω2

v2

)]

Alfven ± Entropy

δA 0 (γ − 1) δSγ

δf ikyvδwx

k2z+k2

y

(

v2A(γ − 1) + c2

)

δSγ

δEx ±i kz

vA

δwx

k2z+k2

y

0

¯δvx ∓ikyvAv

δwx

k2z+k2

y

0

δEyiky

vδwx

k2z+k2

y

0

δK1

(

ωkz + k2yv)

δwx

k2z+k2

y

−iky(

v2A(γ − 1) + c2

)

δSγ

δS 0 δS

kzωv± kxvA

vωv

In a uniform flow, the perturbations can be decomposed into 7 different waves, which are the eigenvectors of thedifferential system: an entropy wave, 2 Alfven waves and 4 magnetosonic waves (2 fast and 2 slow). The perturbationassociated with each of these waves is given in Table A.4, where ωc ≡ ω − kzv is the frequency of a wave in theframe comoving with the fluid, µ2 ≡ 1−k2xc

2(

1−M2)

/ω2, and k±z0 ≡ ω/c× (M∓ µ) /(

1−M2)

is the non-magnetic

MHD SASI 11

wavenumber of the sound waves. The vertical wavevector of the magnetosonic waves is obtained by numerically solvingthe fourth order polynomial dispersion relation:

ω4c − ω2

ck2(

c2 + v2A)

+ k2c2k2xv2A = 0. (A27)

The slow waves are then distinguished from the fast waves as follows: c2 < ω2c/k

2 < c2 + v2A for the fast waves, and0 < ω2

c/k2 < v2A for the slow ones (vA < c in the case considered here). (+) and (-) waves are distinguished by

computing the group velocity ∂ω/∂kz along the z direction. An approximate expression of kz that is valid in the weakfield limit is also given in Table A.4.The expressions of Table A.4 are the coefficients of a matrix which, when multiplied by the following amplitude

vector[

(

δρ

ρ

)f±

,

(

δvxc

)s±

,

(

δwx

k2z + k2y

)a±

, δS

]

, (A28)

gives the corresponding perturbations. This matrix is inversed numerically in order to determine the amplitude ofeach wave as a function of the value of the perturbations.

Boundary conditions

To obtain the boundary conditions at the shock, we use the conservation of energy, momentum, and mass fluxes aswell as the continuity of the magnetic field perpendicular to the shock, and of the electric field parallel to the shock, inthe frame of the perturbed shock. Assuming that the upstream flow is unperturbed, this gives the following boundaryconditions as a function of the shock displacement ∆ζ:

δAsh=0, (A29)

δfsh= iωv1∆ζ

(

1−v2v1

)

, (A30)

δExsh=0, (A31)

¯δvxsh=kxδfshω

+ ikxv2A2

v2∆ζ

(

v22v21

− 1

)

, (A32)

= ikx∆ζ

(

1−v2v1

)[

v1 −v2A2

v2

(

1 +v2v1

)]

, (A33)

δEysh=−iω

v2∆ζ

(

1−v2v1

)

, (A34)

δK1sh=0, (A35)

δSsh

γ=

iω

c2∆ζv1

(

1−v2v1

)2

−∂Φ

∂z

∆ζ

c2

(

1−v2v1

)

. (A36)

Below the potential jump, we use a leaking boundary condition, i.e. no wave propagates upward. For this purpose, atz = z∇−3H∇ where the flow is homogeneous, we decompose the perturbations into waves as described in the previoussection. The only wave that can propagate upward is the fast magnetosonic wave f−. The boundary condition requiresthat its amplitude is zero:

(

δρ

ρ

)f−

= 0. (A37)

Computation of the cycle efficiencies

The coupling efficiencies at the shock are computed by decomposing into waves the perturbations just below theshock. For a given value of ω, these perturbations are set by the upper boundary condition described above. Thecoupling efficiency Qi

sh is then computed as the ratio of the amplitude of the wave i (i = s±, f+, a±, S) to the amplitudeof the upward propagating fast magnetosonic wave:

Qish ≡

δAish

δAf−sh

. (A38)

To obtain the coupling efficiency Qi∇, we need to determine in what proportion an upward propagating fast wave

(f−) must be added to a wave i at the upper boundary at zsh, so that the lower boundary condition is respected. Forthis purpose, we successively set the perturbation at the shock to : (i) a wave i (described in table A.4) of amplitude

δAish = 1, (ii) an upward propagating acoustic wave (f− in table A.4) of amplitude δAf−

sh = 1. We then integrate thedifferential system from the shock to zlow = z∇ − 3H∇, where the lower boundary condition is estimated through the

amplitude of the upward propagating wave f− wave: x ≡(

δρρ

)f−

zlow. This resulting quantity is called xi for the wave

12 Guilet & Foglizzo

for the case (i) and xf− for case (ii). The leaking boundary at zlow is respected for a combination of the waves i andf− (at the shock) if :

δAf−sh x

f− + δAishx

i = 0. (A39)

Thus the coupling efficiency can be computed as :

Qi∇ =

δAf−sh

δAish

= −xi

xf−. (A40)

VERTICAL MAGNETIC FIELD

In this appendix we write the equations governing the perturbations in the case of a vertical magnetic field, usingthe same method as described above in the case of a horizontal magnetic field.

Definition of the new variables

δh ≡ δρρ + δvz

vδρρ = 1

1−M2

[

δfc2 −M2δh− δS

]

δf ≡ vδvz +2cδcγ−1

δvzv = 1

1−M2

[

− δfc2 + δEy + δS

]

δEx ≡ δvx − v δBx

BδBz

B = kx

ω δEx

¯δvx ≡ δvx −v2A

vδBx

B δvx = 1

1−v2A

v2

(

¯δvx −v2A

v2 δEx

)

δEy ≡ δvy − vδBy

BδBx

B = 1

v

(

1−v2A

v2

)

(

¯δvx − δEx

)

¯δvy ≡ δvy −v2A

vδBy

B δvy = 1

1−v2A

v2

(

¯δvy −v2A

v2 δEy

)

δS ≡ 1γ−1

(

δPP − γ δρ

ρ

)

δBy

B = 1

v

(

1−v2A

v2

)

(

¯δvy − δEy

)

Differential system

∂δh

∂z=

iω

v (1−M2)

[

δf

c2−M2δh− δS

]

−ikx

v(

1−v2A

v2

)

[

¯δvx −v2Av2

δEx

]

, (B1)

∂δf

∂z=

iωv

1−M2

[

−δf

c2+ δh+

(

γ − 1 +1

M2

)

δS

γ

]

, (B2)

∂δEx

∂z=

iω

v(

1−v2A

v2

)

[

δEx − ¯δvx]

, (B3)

∂ ¯δvx∂z

=iω

v(

1−v2A

v2

)

[

¯δvx −v2Av2

(

1 +k2x(

v2 − v2A)

ω2

)

δEx

]

+ikxv

1−M2

[

−δf

v2+ δh+

(

γ − 1 +1

M2

)

δS

γ

]

, (B4)

∂δEy

∂z=

iω

v(

1−v2A

v2

)

[

δEy − ¯δvy]

, (B5)

∂ ¯δvy∂z

=iω

v(

1−v2A

v2

)

[

¯δvy −v2Av2

δEy

]

, (B6)

∂δS

∂z=

iω

vδS. (B7)

(B8)

MHD SASI 13

TABLE 2

Decomposition into waves: vertical magnetic field

Fast ± Slow ± Alfven ± Entropy

δh(

1 + c2

v2kzvωc

)

δρρ

kx

ω2c−k2

zc2

(

ωc +c2

v2 kzv)

δvx 0 (1− γ) δSγ

δf c2(

1 + kzvωc

)

δρρ

c2 kx

ω2c−k2

zc2

(ωc + kzv) δvx 0 c2 δSγ

δExωc

kx

(

1− k2zc2

ω2c

)

(

1 + kzvωc

)

δρρ

(

1 + kzvωc

)

δvx 0 0

¯δvxωc

kx

(

1− k2zc2

ω2c

)(

1 +v2A

v2kzvωc

)

δρρ

(

1 +v2A

v2kzvωc

)

δvx 0 0

δEy 0 0(

1± vvA

)

δvya± 0¯δvy 0 0

(

1± vAv

)

δvya± 0δS 0 0 0 δS

kz ∼ ωc

M∓µ

1−M2 ± v2A

2c2k2xc

µω∼ ω

v∓vA± 1

2

v3A

v3ω3

c2v

k2xv2+ω2

ωv∓vA

ωv

Wave decomposition in a uniform flow

Boundary conditions at the shock

δhsh=−iω∆ζ

v2

(

1−v2v1

)

, (B9)

δfsh= iωv1∆ζ

(

1−v2v1

)

, (B10)

δExsh=0, (B11)

¯δvxsh=kxωδfsh = ikxv1∆ζ

(

1−v2v1

)

, (B12)

δEysh=0, (B13)

¯δvysh=0, (B14)

δSsh

γ=

iωv1∆ζ

c2

(

1−v2v1

)2

−∂Φ

∂z

∆ζ

c2

(

1−v2v1

)

. (B15)

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